Is there a log file which keeps track of things users try to do and are denied due to regular unix file permissions. I know selinux does things, but a lot of the time the good ole' file permissions stop them first. When this happens, is there a log that gets printed to.

You might be able to find auditing software that may have enhanced auditing or maybe software (like SeLinux) that uses more complex ACL access to files and it may log things, but even Windows doesn't log such permission errors (there are utilities with Sysinternals that show on-the-fly access denied errors for Windows, though).

I don't think I've even run across any utilities similar that that functionality in Unix systems.

You can try looking for "incidental" things in logs, like mail or web server programs that log errors trying to access particular file paths that as an administrator you know should be there.

If you're interested in security on your system to keep pesky users from acting pesky, you can try some of the utilities listed here and see if they will help you.